Octic reciprocity

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Thompson's Construction Algorithm (TCA) derives a nondeterministic finite automaton (NFA) from any regular expression by splitting it into its constituent subexpressions, from which the NFA will be constructed using a set of rules.

It was invented by Ken Thompson.[1]

Rules

The following rules are depicted according to Aho et. al (1986),[2] p. 122. N(s) and N(t) are NFAs of the subexpressions s and t respectively.

The expression ε is converted to

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A symbol a of the input alphabet is converted to

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The union expression s|t is converted to

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State q goes via ε either to the initial state of N(s) or N(t). Their final states become intermediate states of the whole NFA and merge via two ε-transitions into the final state of the NFA.

The concatenation expression st is converted to

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The initial state of N(s) is the initial state of the whole NFA. The final state of N(s) becomes the initial state of N(t). The final state of N(t) is the final state of the whole NFA.

The Kleene star expression s* is converted to

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An ε-transition connects initial and final state of the NFA with the sub-NFA N(s) in between. Another ε-transition from the inner final to the inner initial state of N(s) allows for repetition of expression s according to the star operator.

  • The parenthesized expression (s) is converted to N(s) itself.

References

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  1. Programming Techniques: Regular expression search algorithm
  2. Alfred V. Aho, Ravi Sethi, Jeffrey Ullman: Compilers: Principles, Techniques and Tools. Addison Wesley, 1986